If `xy(x + y) = 1` then,the value of `1/(x^3 y^3) -x^3 - y^3` is

Options

A. 3
B. `-3`
C. 1
D. `-1`

Answer

To solve the problem, we start with the equation given:

**Step 1: Start with the equation**
Given:
\[ xy(x + y) = 1 \]

**Step 2: Express \( x + y \) in terms of \( xy \)**
From the equation, we can express \( x + y \):
\[ x + y = \frac{1}{xy} \]

**Hint for Step 2:** Remember that if you have a product equal to a constant, you can isolate one of the terms.

**Step 3: Cube both sides**
Now, we will cube both sides:
\[ (x + y)^3 = \left(\frac{1}{xy}\right)^3 \]

**Step 4: Expand the left-hand side using the binomial theorem**
Using the binomial expansion:
\[ (x + y)^3 = x^3 + y^3 + 3xy(x + y) \]
Substituting \( x + y \):
\[ (x + y)^3 = x^3 + y^3 + 3xy\left(\frac{1}{xy}\right) \]
This simplifies to:
\[ (x + y)^3 = x^3 + y^3 + 3 \]

**Hint for Step 4:** When expanding cubes, remember to include all terms from the binomial expansion.

**Step 5: Substitute the right-hand side**
Now substituting the right-hand side:
\[ \left(\frac{1}{xy}\right)^3 = \frac{1}{x^3y^3} \]
So we have:
\[ x^3 + y^3 + 3 = \frac{1}{x^3y^3} \]

**Hint for Step 5:** Be careful with fractions; ensure you maintain the correct operations when substituting.

**Step 6: Rearranging the equation**
Rearranging gives us:
\[ x^3 + y^3 = \frac{1}{x^3y^3} - 3 \]

**Step 7: Finding the required expression**
Now, we need to find:
\[ \frac{1}{x^3y^3} - x^3 - y^3 \]
Substituting from the previous result:
\[ \frac{1}{x^3y^3} - (x^3 + y^3) = \frac{1}{x^3y^3} - \left(\frac{1}{x^3y^3} - 3\right) \]

**Step 8: Simplifying the expression**
This simplifies to:
\[ \frac{1}{x^3y^3} - \frac{1}{x^3y^3} + 3 = 3 \]

**Final Answer:**
Thus, the value of \( \frac{1}{x^3y^3} - x^3 - y^3 \) is:
\[ \boxed{3} \]